Subject: Factor Theorem

Name: Yael

Who is asking: Student Level: Secondary

Question: Hi, I'm not sure what to do with this question:

Prove (x - a - b) is a factor of x3 - a3 - b3 - 3ab (a + b)

I don't know even how to start it!

Thanks



Hi Yael,

The integer 3 is a factor of 414. I know this because, if I divide 414 by 3 the remainder is zero. You could apply the same argument here. Divide x3 - a3 - b3 - 3ab (a + b) by (x -a -b) and show that the remainder is zero. Fortunately there is another procedure you can apply.

The factor theorem states that

If f(x) is a polynomial then (x - p) is a factor of f(x) if and only if f(p) = 0. In your example the polynomial is f(x) = x3 - a3 - b3 - 3ab (a + b) and the expression that you wish to show is a factor is (x - a - b) = [x - (a + b)] Hence you can show that (x - a - b) is a factor of f(x) by showing that f(a + b) = 0.

f(a + b) = (a + b)3 - a3 - b3 - 3ab (a + b)

Expand (a + b)3 and show that f(a + b) = 0.

Penny

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